Abstract
Performancebased earthquake engineering offers a versatile framework for quantifying the seismic performance of structures. Its implementation requires a comprehensive description of the nonlinear structural behavior, facilitated typically via multiple nonlinear response history analyses (NLRHAs). This burden can be very high when highfidelity finite element models (FEMs) are used to describe structural response. To alleviate it, approximations are commonly employed, using a moderate number of analyses, or even replacing altogether the NLRHA with a nonlinear static analysis. This contribution explores two alternative paths for accommodating the desired computational efficiency: (a) use of reduced order models that are calibrated to closely match the original FEM; (b) adoption of multifidelity Monte Carlo (MFMC) that combines the original FEM to guarantee unbiased predictions and the aforementioned reduced order models to accelerate the Monte Carlo implementation. Advancements are established for the MFMC implementation, in order to accommodate the efficient propagation of the different sources of uncertainty across the estimation of the different seismic performance statistics of interest. The accuracy and computational benefits are illustrated for two benchmark structures over two different output variables: repair cost (resiliency quantification) and embodied energy associated with repairs (sustainability quantification).
Similar content being viewed by others
Data availability
The data used to support the findings of this study are available from the corresponding author upon request.
References
Angeles K, Patsialis D, Taflanidis AA, KijewskiCorrea TL, Buccellato A, Vardeman C (2021) Advancing the design of resilient and sustainable buildings: an integrated lifecycle analysis. J Struct Eng 147(3):04020341. https://doi.org/10.1061/(ASCE)ST.1943541X.0002910
ASCE/SEI (2016) Minimum design loads for buildings and other structures. ASCE/SEI 7–16, Reston, VA
Bai JW, Hueste MBD, Gardoni P (2009) Probabilistic assessment of structural damage due to earthquakes for buildings in MidAmerica. J Struct Eng 135(10):1155–1163
Baker JW (2011) Conditional mean spectrum: tool for groundmotion selection. J Struct Eng 137(3):322–331
Baker JW, Cornell CA (2008) Uncertainty propagation in probabilistic seismic loss estimation. Struct Saf 30(3):236–252
Baltzopoulos G, Baraschino R, Iervolino I (2019) On the number of records for structural risk estimation in PBEE. Earthquake Eng Struct Dynam 48(5):489–506
Bazzurro P, Cornell CA, Shome N, Carballo JE (1998) Three proposals for characterizing MDOF nonlinear seismic response. J Struct Eng 124(11):1281–1289
Bozorgnia Y, Bertero VV (2004) Earthquake engineering: from engineering seismology to performancebased engineering. CRC Press
Bradley BA, Lee DS (2010) Component correlations in structurespecific seismic loss estimation. Earthq Eng Struct Dyn 39(3):237–258
Cha EJ, Ellingwood BR (2013) Seismic risk mitigation of building structures: the role of risk aversion. Struct Saf 40:11–19
Chang DY (1993) Parsimonious modeling of inelastic structures. September 30California Institute of Technology,
Cornell C, Krawinkler H (2000) Progress and challenges in seismic performance assessment. PEER Center News 3. University of California, Berkeley
Der Kiureghian AD (2005) Nonergodicity and PEER’s framework formula. Earthq Eng Struct Dyn 34(13):1643–1652
FEMA440 (2005) Improvement of nonlinear static seismic analysis procedures. FEMA440, Redwood City 7 (9):11
FEMAP583.1 (2012) Seismic performance assessment of buildings, Volume 3Performance assessment calculation tool (PACT). Federal Emergency Management Agency Redwood City, CA
FEMAP58 (2012) Seismic performance assessment of buildings. American Technology Council, Redwood City, CA
Filippou FC, Bertero VV, Popov EP (1983) Effects of bond deterioration on hysteretic behavior of reinforced concrete joints
Fragiadakis M, Lagaros ND, Papadrakakis M (2006) Performancebased multiobjective optimum design of steel structures considering lifecycle cost. Struct Multidiscip Optim 32(1):1
Frankel A, Leyendecker E (2001) Seismic hazard curves and uniform hazard response spectra for the United States. OpenFile Report:01436
Gehl P, Douglas J, Seyedi DM (2015) Influence of the number of dynamic analyses on the accuracy of structural response estimates. Earthq Spectra 31(1):97–113
Gencturk B, Hossain K, Lahourpour S (2016) Life cycle sustainability assessment of RC buildings in seismic regions. Eng Struct 110:347–362
Gentile R, Galasso C (2021) Simplicity versus accuracy tradeoff in estimating seismic fragility of existing reinforced concrete buildings. Soil Dyn Earthq Eng 144:106678
Gidaris I, Taflanidis AA (2015) Performance assessment and optimization of fluid viscous dampers through lifecycle cost criteria and comparison to alternative design approaches. Bull Earthq Eng 13(4):1003–1028
Gidaris I, Taflanidis AA, Mavroeidis GP (2018) Multiobjective design of supplemental seismic protective devices utilizing lifecycle performance criteria. J Struct Eng 144(3):04017225
Goulet CA, Haselton CB, MitraniReiser J, Beck JL, Deierlein G, Porter KA, Stewart JP (2007) Evaluation of the seismic performance of codeconforming reinforcedconcrete frame buildingFrom seismic hazard to collapse safety and economic losses. Earthq Eng Struct Dynam 36(13):1973–1997
Hammond G, Jones C (2008) Inventory of carbon & energy: ICE, vol 5. Bath: Sustainable Energy Research Team, Department of Mechanical Engineering. University of Bath, UK
Hancock J, Bommer JJ, Stafford PJ (2008) Numbers of scaled and matched accelerograms required for inelastic dynamic analyses. Earthq Eng Struct Dyn 37(14):1585–1607
Haselton CB, Goulet CA, MitraniReiser J, Beck JL, Deierlein GG, Porter KA, Stewart JP, Taciroglu E (2008) An assessment to benchmark the seismic performance of a codeconforming reinforcedconcrete momentframe building. Pacific Earthquake Engineering Research Center (2007/1)
Hasik V, Chhabra JP, Warn GP, Bilec MM (2018) Review of approaches for integrating loss estimation and life cycle assessment to assess impacts of seismic building damage and repair. Eng Struct 175:123–137
HAZUS (2003) Multihazard Loss Estimation Methodology. National Institute of Bulding Sciences and Federal Emergency Management Agency (NIBS and FEMA), Federal Emergency Management Agency, Washington DC
Hisham M, Yassin M (1994) Nonlinear analysis of prestressed concrete structures under monotonic and cycling loads. University of California, Berkeley Ph D thesis
Hofer L, Zanini MA, Faleschini F, Pellegrino C (2018) Profitability analysis for assessing the optimal seismic retrofit strategy of industrial productive processes with businessinterruption consequences. J Struct Eng 144(2):04017205
Huang YN, Whittaker AS, Luco N, Hamburger RO (2011) Scaling earthquake ground motions for performancebased assessment of buildings. J Struct Eng 137(3):311–321
Iervolino I (2017) Assessing uncertainty in estimation of seismic response for PBEE. Earthq Eng Struct Dynam 46(10):1711–1723
Iervolino I, Giorgio M, Polidoro B (2014) Sequencebased probabilistic seismic hazard analysis. Bull Seismol Soc Am 104(2):1006–1012
Ismail M, Ikhouane F, Rodellar J (2009) The hysteresis BoucWen model, a survey. Arch Comput Methods Eng 16(2):161–188
Jalayer F, Beck J (2008) Effects of two alternative representations of groundmotion uncertainty on probabilistic seismic demand assessment of structures. Earthq Eng Struct Dyn 37(1):61–79
Jalayer F, Beck J, Zareian F (2012) Analyzing the sufficiency of alternative scalar and vector intensity measures of ground shaking based on information theory. J Eng Mech 138(3):307–316
Katsanos E, Sextos A, Elnashai AS (2014) Prediction of inelastic response periods of buildings based on intensity measures and analytical model parameters. Eng Struct 71:161–177
Kazantzi A, Vamvatsikos D (2021) Practical performancebased design of friction pendulum bearings for a seismically isolated steel top story spanning two RC towers. Bull Earthq Eng 19(2):1231–1248
Kohrangi M, Bazzurro P, Vamvatsikos D (2016) Vector and scalar IMs in structural response estimation, Part I: Hazard analysis. Earthq Spectra 32(3):1507–1524
Kostic SM, Filippou FC (2011) Section discretization of fiber beamcolumn elements for cyclic inelastic response. J Struct Eng 138(5):592–601
Krawinkler H, Seneviratna G (1998) Pros and cons of a pushover analysis of seismic performance evaluation. Eng Struct 20(4–6):452–464
Kyprioti AP, Taflanidis AA (2021) Kriging metamodeling for seismic response distribution estimation. Earthq Eng Struct Dyn 50(13):3550–3576
Lagaros ND, Magoula E (2013) Lifecycle cost assessment of midrise and highrise steel and steel–reinforced concrete composite minimum cost building designs. Struct Des Tall Spec Build 22(12):954–974
Lamprou A, Jia G, Taflanidis AA (2013) Lifecycle seismic loss estimation and global sensitivity analysis based on stochastic ground motion modeling. Eng Struct 54:192–206
Lignos DG, Putman C, Krawinkler H (2015) Application of simplified analysis procedures for performancebased earthquake evaluation of steel special moment frames. Earthq Spectra 31(4):1949–1968
Lin T, Haselton CB, Baker JW (2013) Conditional spectrumbased ground motion selection. Part I: hazard consistency for riskbased assessments. Earthq Eng Struct Dyn 42(12):1847–1865
Loh CH, Jean WY, Penzien J (1994) Uniformhazard response spectra—an alternative approach. Earthquake Eng Struct Dynam 23(4):433–445
McGuire RK (1995) Probabilistic seismic hazard analysis and design earthquakes: closing the loop. Bull Seismol Soc Am 85(5):1275–1284
McKenna F (2011) OpenSees: a framework for earthquake engineering simulation. Comput Sci Eng 13(4):58–66
MitraniReiser J (2007) An ounce of prevention: probabilistic loss estimation for performancebased earthquake engineering. California Institute of Technology
Moehle J, Deierlein GG (2004) A framework methodology for performancebased earthquake engineering. In: 13th world conference on earthquake engineering
Munjy H, Zareian F (2018) Efficient statistical approximation of engineering demand parameters in building structures. 16th European Conference on Earthquake Engineering
Nettis A, Gentile R, Raffaele D, Uva G, Galasso C (2021) Cloud capacity spectrum method: accounting for recordtorecord variability in fragility analysis using nonlinear static procedures. Soil Dyn Earthq Eng 150:106829
Ohtori Y, Christenson R, Spencer B Jr, Dyke S (2004) Benchmark control problems for seismically excited nonlinear buildings. J Eng Mech 130(4):366–385
Patsialis D, Kyprioti AP, Taflanidis AA (2020) Bayesian calibration of hysteretic reduced order structural models for earthquake engineering applications. Eng Struct 224:111204
Patsialis D, Taflanidis A (2021) Multifidelity Monte Carlo for seismic risk assessment applications. Struct Saf 93:102129
Patsialis D, Taflanidis AA (2020) Reduced order modeling of hysteretic structural response and applications to seismic risk assessment. Eng Struct 209:110135. https://doi.org/10.1016/j.engstruct.2019.110135
PEER (2013/03) PEER NGAWEST2 Database (Pacific Earthquake Engineering Research Center (PEER), California). https://ngawest2.berkeley.edu/users/sign_in
Peherstorfer B, Beran PS, Willcox KE (2018) Multifidelity Monte Carlo estimation for largescale uncertainty propagation. In: 2018 AIAA NonDeterministic Approaches Conference. p 1660
Peherstorfer B, Willcox K, Gunzburger M (2016) Optimal model management for multifidelity Monte Carlo estimation. SIAM J Sci Comput 38(5):A3163–A3194
Porter K, Ramer K (2012) Estimating earthquakeinduced failure probability and downtime of critical facilities. J Bus Contin Emer Plan 5(4):352–364
Porter KA, Kiremidjian AS, LeGrue JS (2001) Assemblybased vulnerability of buildings and its use in performance evaluation. Earthq Spectra 17(2):291–312
Poulos A, de la Llera JC, MitraniReiser J (2017) Earthquake risk assessment of buildings accounting for human evacuation. Earthq Eng Struct Dyn 46(4):561–583
Reyes JC, Kalkan E (2012) How many records should be used in an ASCE/SEI7 ground motion scaling procedure? Earthq Spectra 28(3):1223–1242
Sevieri G, Gentile R, Galasso C (2021) A multifidelity Bayesian framework for robust seismic fragility analysis. Earthq Eng Struct Dyn 50(15):4199–4219
Shin H, Singh M (2014) Minimum failure costbased energy dissipation system designs for buildings in three seismic regions–Part II: Application to viscous dampers. Eng Struct 74:275–282
Spillatura A, Kohrangi M, Bazzurro P, Vamvatsikos D (2021) Conditional spectrum record selection faithful to causative earthquake parameter distributions. Earthq Eng Struct Dyn
Sullivan TJ, Welch DP, Calvi GM (2014) Simplified seismic performance assessment and implications for seismic design. Earthq Eng Eng Vib 13(1):95–122
Taflanidis AA, Beck JL (2009) Lifecycle cost optimal design of passive dissipative devices. Struct Saf 31(6):508–522
USGS (2022) United States geological survey national seismic hazard mapping project application programming interface. http://usgs.github.io/nshmphaz/javadoc/. Accessed 1 January 2022
Vamvatsikos D (2013) Derivation of new SAC/FEMA performance evaluation solutions with secondorder hazard approximation. Earthq Eng Struct Dynam 42(8):1171–1188
Vamvatsikos D (2014) Seismic performance uncertainty estimation via IDA with progressive accelerogramwise latin hypercube sampling. J Struct Eng 140(8):A4014015
Vamvatsikos D, Allin Cornell C (2006) Direct estimation of the seismic demand and capacity of oscillators with multilinear static pushovers through IDA. Earthq Eng Struct Dynam 35(9):1097–1117
Vamvatsikos D, Fragiadakis M (2010) Incremental dynamic analysis for estimating seismic performance sensitivity and uncertainty. Earthq Eng Struct Dyn 39(2):141–163
Vamvatsikos D, Kazantzi AK, Aschheim MA (2016) Performancebased seismic design: avantgarde and codecompatible approaches. ASCEASME J Risk Uncertain Eng Syst Part a: Civ Eng 2(2):C4015008
Vitiello U, Asprone D, Di Ludovico M, Prota A (2017) Lifecycle cost optimization of the seismic retrofit of existing RC structures. Bull Earthq Eng 15(5):2245–2271
Wei HH, Shohet IM, Skibniewski MJ, Shapira S, Yao X (2016) Assessing the lifecycle sustainability costs and benefits of seismic mitigation designs for buildings. J Archit Eng 22(1):04015011
Funding
Not applicable.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare there are no conflicts of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: Computational details for estimation of output variable statistics conditional on EDP
This appendix discusses the estimation of conditional on EDP statistics for OVs, addressing computational details for the Consequence Module. Following the definitions in Sect. 2.3, assume that for the ith damageable assembly \(n_{di}\), different damage states \(d_{ik} , \, k = 0, \ldots ,n_{di}\) are designated, with damage state for k = 0 corresponding to undamaged conditions. The fragility of the ith damageable assembly is related to engineering demand parameter \(EDP_{i}\). The probability that the damage measure related to the ith assembly (DM_{i}) exceeds damage state d_{ik} is quantified through its respective fragility function. When, as customary, a lognormal distribution is utilized for these functions, the resultant expression is:
where Φ denotes the standard Gaussian cumulative distribution function, and \(\overline{b}_{ik}\) and β_{ik} are the median and logarithmic standard deviation for the fragility function. This fragility quantification \(P(DM_{i} > d_{ik} EDP_{i} = edp_{i} )\) can be equivalently interpreted to correspond to \(P(b_{ik} < edp_{i} )\), with \(b_{ik}\) defined as the random variable for the threshold associated with damage state d_{ik}, which for the case of the fragility function of Eq. (18) follows a lognormal distribution with median \(\overline{b}_{ik}\) and logarithmic standard deviation β_{ik}. This shows that the fragility quantification may be equivalently considered to define a probability model for the threshold \(b_{ik}\) (treated as a random variable) defining damage state d_{ik}, with exceedance of the damage state corresponding to \(edp_{i} > b_{ik}\) and undamaged conditions corresponding to \(edp_{i} \le b_{i1}\). The probabilistic quantification of losses for each damageable assembly is accommodated through a probabilistic description for v_{ik}, the losses associated with the kth damage state for the ith assembly.
Let b and v denote vectors defined by augmenting all elements of \(\{ b_{ik} {; }k = 1, \ldots ,n_{di} , \, i = 1,...,n_{as} \}\) and \(\{ v_{ik} {; }k = 1, \ldots ,n_{di} , \, i = 1,...,n_{as} \}\), respectively. The fragility quantification for each \(b_{ik}\) as well as the correlation across all elements of b defines ultimately the joint probability model p(b) for b, and similarly, the probabilistic description for v_{ik} and the correlation across damageable assemblies defines the joint probability model p(v) for v. According to Eq. (4), and using sample set \(\{ {\mathbf{z}}^{im(s)} {; }s = 1, \ldots ,N_{g}^{im} \}\) or its parametric fit \(LN({{\varvec{\upmu}}}^{im} ,{{\varvec{\Sigma}}}^{im} )\) as discussed in Sect. 2.3, independent samples \(OVIM = im\) as well as samples for the losses associated with each damageable assembly \(OV_{i} IM = im\) can be generated according to the following process, termed herein as conditional OV sampling:

Step 1: Generate an EDPIM = im sample \({\mathbf{z}}^{im(r)}.\) This is done either by sampling (with replacement) one sample from set \(\{ {\mathbf{z}}^{im(s)} {; }s = 1, \ldots ,N_{g}^{im} \}\) or, if parametric fit is employed, generating a sample from the fitted distribution \(LN({{\varvec{\upmu}}}^{im} ,{{\varvec{\Sigma}}}^{im} )\). These two variant implementations will be distinguished herein using terminology, EDP resampling and EDP approximation, respectively.

Step 2: Generate a sample \({\mathbf{b}}^{(r)}\) from p(b) and a sample \({\mathbf{v}}^{(r)}\) from p(v).

Step 3: For the ith damageable assembly, compare the corresponding edp_{i} value (for the ith component of EDP vector) from sample \({\mathbf{z}}^{im(r)}\) to the thresholds from sample \({\mathbf{b}}^{(r)}\) associated with the damaged states (of the assembly), and determine in which damage state the assembly belongs to (edp_{i} value exceeds the threshold for this damage state but does not exceed the threshold for the following one). The losses for the assembly are defined by the respective index of \({\mathbf{v}}^{(r)}\), which ultimately provides sample \(ov_{i}^{im(r)}\) from \(OV_{i} IM = im\). Repeat this for all assemblies.

Step 4: Combine \(ov_{i}^{(r)}\) samples to obtain sample \(ov_{{}}^{im(r)}\) from \(OVIM = im\).
Utilizing a sufficient number of samples \(\{ ov_{{}}^{im(r)} ;r = 1,...,N_{r} \}\) the statistics for OV as well as the complementary cumulative distribution function can be estimated:
where I[\(\cdot\)] represents the indicator function, corresponding to one if the relationship inside the brackets holds and to 0 else. Similar relationships hold for the statistics for OV_{i}.
If only the expected value of Eq. (19) is warranted, then an analytical approximation can be efficiently implemented. This is accommodated by first calculating
where n_{i} is the number of identical elements in the ith assembly,\(\overline{v}_{ik}\) is the mean of v_{ik}, and, assuming perfect correlation (Porter et al. 2001; FEMAP58 2012) between the damage states, the conditional probability of being in each damage state is estimated as:
This then leads to:
Utilizing this analytical approach, Steps 2–4 in the conditional OV sampling are replaced by using the average of the quantity presented in Eq. (24) over the EDP samples obtained in Step 1. If the lognormal fit is used for the EDP distribution, then even Step 1 can be integrated within the analytical estimation (Baker and Cornell 2008; Angeles et al. 2021). Similar extensions to estimate \(Var[OVEDP = edp]\) or higher order statistics through analytical or semianalytical approximations involve significant computational burden for properly capturing correlations between damage states and damageable assemblies (Angeles et al. 2021). This approach is not recommended for applications with large number of damage states and assemblies. Note though that for the variance for individual OV_{i}’s, \(Var[OV_{i} EDP_{i} = edp_{i} ]\), a simple analytical expression similar to Eq. (22) can be obtained (Angeles et al. 2021), whereas estimation of \(Var[OVEDP = edp]\) can be efficiently analytically performed when correlations across assemblies are ignored (Angeles et al. 2021).
Appendix 2: Response approximation using nonlinear static analysis
This appendix reviews the approximation of the distribution \(EDPIM\sim LN({{\varvec{\upmu}}}^{im} ,{{\varvec{\Sigma}}}^{im} )\) through static nonlinear analysis of FEMA P58 (FEMAP58 2012) for low/mediumrise buildings (below 15 stories, with small higher mode contributions), with insignificant Pdelta effects (drift ratios lower than 4%) and no irregularities in plan and elevation. Note that many advanced variants exist in the literature for improving loss estimation using nonlinear static analysis, for example (Nettis et al. 2021), and the simplified (in comparison) approach adopted here is chosen simply as a popular, and promoted in FEMA P58 formulation. According to it, fundamental periods and mode shapes of the building are used to compute the vertical distribution of equivalent static forces with (total) base shear for IM = im given by \(V = C_{1} C_{2} S_{a} (T_{1} )W_{1}\) (FEMAP58 2012) where C_{1} is an adjustment factor for inelastic displacements (function of soil condition and yield base shear), C_{2} is an adjustment factor for cyclic degradation (function of soil condition and yield base shear), S_{a}(T_{1}) is the 5% damped mean spectrum of the ground motion for the fundamental period of the building corresponding to the utilized IM, and W_{1} is the effective modal weight for the first mode (not less than 80% of the total building weight). The yield base shear is calculated through nonlinear static pushover analysis based on the building FEM, adopting a bilinearization based on the target displacement according to FEMA440 standard (FEMA440 2005). Shear V is distributed along the height of the building using equivalent static force distributions and elastic drift demands are then calculated through linear analysis. For each IM level, this information, along with the estimated peak ground acceleration, is used to compute the median estimates for the different engineering demand parameters, to accommodate the calculation of \({{\varvec{\upmu}}}^{im}\). More specifically, from peak ground acceleration, the peak floor accelerations are estimated, along with the peak ground velocity that is then used to estimate the peak floor velocities. Inelastic behavior is accounted for through an adjustment factor that depends on the relationship between the base shear per IM and the yield base shear calculated through the nonlinear pushover analysis (FEMAP58 2012). For the approximation of \({{\varvec{\Sigma}}}^{im}\), the dispersion for the median EDP predictions due to record to record variability is obtained from Table 5–6 of FEMA P581 Methodology (FEMAP58 2012). Note that this dispersion differs for each IM since it depends on S_{a}(T_{1}). The complete covariance matrix \({{\varvec{\Sigma}}}^{im}\), needed to estimate higherorder OV statistics, is obtained by assuming, additionally, perfect correlation for the EDP vector.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author selfarchiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Patsialis, D., Taflanidis, A.A. & Vamvatsikos, D. Improving the computational efficiency of seismic buildingperformance assessment through reduced order modeling and multifidelity Monte Carlo techniques. Bull Earthquake Eng 21, 811–847 (2023). https://doi.org/10.1007/s10518022015514
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10518022015514